Given $ \overrightarrow{PQ}\perp\overrightarrow{PS}$, $ m \angle QPR = 4x - 18$, and $ m \angle RPS = 7x + 9$, find $m\angle QPR$. $P$ $Q$ $S$ $R$
Solution: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Since we are given that $\overrightarrow{PQ}\perp\overrightarrow{PS}$ , we know ${m\angle QPS = 90}$ Substitute in the expressions that were given for each measure: $ {4x - 18} + {7x + 9} = {90}$ Combine like terms: $ 11x - 9 = 90$ Add $9$ to both sides: $ 11x = 99$ Divide both sides by $11$ to find $x$ $ x = 9$ Substitute $9$ for $x$ in the expression that was given for $m\angle QPR$ $ m\angle QPR = 4({9}) - 18$ Simplify: $ {m\angle QPR = 36 - 18}$ So ${m\angle QPR = 18}$.